This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for (...) publication. Using manuscripts from his Nachlass, we exhibit essential features of these extensions. (shrink)

Using hitherto unpublished manuscripts from the Zermelo Nachlass, I describe the development of the notion of definiteness and the discussion about it, giving a conclusive picture of Zermelo's thoughts up to the late thirties. As it turns out, Zermelo's considerations about definiteness are intimately related to his concept of a Cantorian universe of categorically definable sets that may be considered an inner model of set theory in an ideationally given universe of classes.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...) of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

Christliche Ethik in der technischen Welt.--Kybernetik und christliches Ethos.--Christliche Anthropologie und biologische Zukunft des Menschen.--Die Stadt als Ort der Utopie.--Christliche Zukunftshoffnung und rationale Zukunftsplanung.--Verantwortung für die Folgen der Freiheit, theologische Überlegungen zum Thema Umweltschutz.

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan `open sets are semidecidable properties'. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical Rice-Shapiro (...) Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers. (shrink)

Was Interkulturelle Philosophie ist, und worin ihr besonderer Beitrag zur Philosophie im ganzen bestehen kann, das kann sich allein in konkreten Ausarbeitungen erweisen. Der vorliegende Band repräsentiert eine breitgefächerte Auswahl sowohl der unterschiedlichen Ansatzpunkte, von denen interkulturelles Philosophieren ausgeht, als auch der diversen Forschungsgebiete innerhalb der Philosophie, die durch interkulturelle Fragestellungen angesprochen sind. Die Vielfalt der historischen und systematischen Zugangsweisen reflektiert die Überzeugung, daß Interkulturelle Philosophie kein in sich geschlossenes Lehrgebäude ist und ein solches auch nicht werden kann, sondern interkulturelles (...) Philosophieren eine bestimmte Einstellung und eine Haltung darbietet, die auf der grundlegenden Annahme beruht, daß die letzte philosophische Einsicht, _die_ Wahrheit im Singular, von niemandem allein erlangt worden ist und auch nicht erlangt werden kann. (shrink)

Die Arbeit behandelt das Verhältnis von Beobachtungs- und theoretischer Sprache anhand der Typenlogik und der Erkenntnistheorie Bertrand Russells. Seine Typentheorie wird auf seine Erkenntnistheorie angewandt. Seine Typentheorie macht deutlich, warum schon aus formalen Gründen eine theoriefreie Beobachtungssprache unmöglich ist.

According to Chen's Linguistic Savings Hypothesis, our native language affects our economic behavior. We present three studies investigating how cross-linguistic differences in the grammaticalization of future-time reference affect intertemporal choices. In a series of decision scenarios about finance and health issues, we let speakers of altogether five languages that represent FTR with increasing strength, that is, Chinese, German, Danish, Spanish, and English, choose between hypothetical sooner-smaller and later-larger reward options. While the LSH predicts a present-bias that increases with FTR-strength, our (...) decision makers preferred later-larger options and this future-bias increased with FTR-strength. In multiple regressions, the FTR-strength effect persisted when controlled for socioeconomic and cultural differences. We discuss why our findings deviate from the LSH and ask in how far the FTR-strength effect represents a habitual constitution of linguistic relativity or an instance of online decision framing. (shrink)

Dieser Band ist Harald Holz zum 65. Geburtstag gewidmet. Er bietet keine Festschrift in dem Sinne, dass in den versammelten Texten an das Werk Harald Holz' im einzelnen angeschlossen wurde. Die Beitrager - Freunde, Schuler, Weggefahrten, Kollegen - mochten vielmehr ihre Anerkennung im allgemeinen dadurch ausdrucken, dass sie sich um die gemeinsame Sache, die Philosophie, bemuhen. In den Texten sind Aspekte der Arbeitsgebiete der einzelnen Autoren mit Interessen des Jubilars verbunden, was sich um so leichter ergab, als letztere ausserordentlich weit (...) gespannt sind. Die Beitrage sind geordnet unter den Titeln: Anschauung und Wesen - Zweifel und Wissen - Normen und Kriterien - Glauben und Handeln.". (shrink)

Carl Friedrich von Weizsäckers Philosophie der Physik kann als eine Reformulierung von Schellings Identitätsphilosophie angesehen werden, indem aber nicht eine ins Absolute geweitete transzendentale Subjektivität den Ausgangpunkt bildet, sondern eine reformulierte Quantentheorie. So wie Schelling in immer erneuten Anläufen versuchte, die Naturwissenschaft seiner Zeit mit seinem geistphilosphischen Ansatz zur Deckung zu bringen, so durchlief von Weizsäcker in seinen Schriften immer wieder den Weg vom Menschen zur Natur und von der Naturwissenschaft zum Menschen. Am Beispiel des Begriffs der »Zeit« als Geschichtszeit (...) wird gezeigt, dass diese Versuche interessante Vermittlungsfiguren hervorbringen, dann aber scheitern, wenn die Zeit der Natur und der menschlichen Geschichtserfahrung identifiziert werden sollen. Solche Versuche sind zudem theologisch fragwürdig, da sie die Transzendenz Gottes ausschließen. (shrink)

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X (...) intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset. (shrink)

How have the theories of aesthetics which were worked out in europe evolved in america? are there widely differing standpoints between european and american aestheticians? what herrmann tried to do, to shed light on these questions, was to look over the issues of "the journal of aesthetics and art criticism" since 1941. thomas munro, a pupil of john dewey and founder of the journal tried to provide in the united states a broader and more open-ended and undogmatic platform for aesthetics (...) including such separate disciplines as philosophy, psychology, and cultural history. this interdisciplinary approach was an important part of his campaign against an a priori aesthetics. once this encyclopedic view concerning art and its theoretical understanding is adopted, one is in a position to go beyond the one-sided, partial, and dogmatic attitude in aesthetics that often dominates in european journals. the author concludes by considering an example that is closely related to contemporary aesthetics as well as to the present american scene. what is the reason that there have been so few articles on these contemporary art movements? (shrink)

The dignity of human embryos. On the moral relevance of potentiality and numerical identity. -.

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T 0 -spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a (...) totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets. (shrink)

Das Buch wendet sich fachubergreifend an Historiker, Staatsrechtler, Philosophen, Politologen und Theologen. Das politisch-philosophische Werk von Hobbes wird als Teil der komplexen Debatten der Revolutionsjahre 1640-1660 gesehen. Die Analyse ist bestrebt, Hobbes' Haltung gegenuber den aktuellen politischen Problemen seiner Zeit aufzudecken, um einen neuen Zugang zu seinen zentralen politischen Aussagen und ihrer Ausformung zu gewinnen. Sie versucht daruber hinaus, durch das Heranziehen von zeitgenossischen Quellen die Rezeptionschancen fur das Hobbes'sche Werk auszuloten und Ruckwirkungen auf das praktische Verhalten und die Theorie (...) von Hobbes zu erhellen. (shrink)